How many pairs of integers (x,y) satisfy the conditions y=3x−1, 6x−2y=2 and 0≤x≤25?
26
By substitution method, we find that there is an integer y for every value of x from 0 to 26
To determine the number of pairs of integers (x, y) that satisfy the given conditions, we need to solve the system of equations.
First, let's solve the two equations:
1. y = 3x - 1
2. 6x - 2y = 2
We can solve this system by substituting the value of y from equation 1 into equation 2:
6x - 2(3x - 1) = 2
6x - 6x + 2 = 2
2 = 2
The equation 2 = 2 is always true, which means the two equations are dependent. This indicates that there are infinitely many solutions that satisfy the conditions.
Now, to find the number of pairs of integers (x, y) that satisfy the conditions, we need to determine the possible values of x within the given range of 0 ≤ x ≤ 25.
We can see that any integer value of x within this range will satisfy the equation y = 3x - 1. Therefore, for every integer value of x, there will be a corresponding value of y that satisfies the equation.
Since there are infinitely many integer values of x within the given range, there will be infinitely many pairs of integers (x, y) that satisfy the conditions.
In conclusion, the number of pairs of integers (x, y) that satisfy the given conditions is infinite.